ANOMALOUS DIFFUSION FOR MULTI-DIMENSIONAL CRITICAL KINETIC FOKKER-PLANCK EQUATIONS

• 成果类型: Article
• 署名作者: Fournier, Nicolas; Tardif, Camille
• 署名单位: Universite Paris Cite; Sorbonne Universite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/20-AOP1426
• 发表日期: 2020
• 页码: 2359-2403
• 关键词: limit; approximation; distributions; functionals; time

摘要

We consider a particle moving in d >= 2 dimensions, its velocity being a reversible diffusion process, with identity diffusion coefficient, of which the invariant measure behaves, roughly, like (1 + vertical bar v vertical bar)(-beta) as vertical bar v vertical bar -> infinity, for some constant beta > 0. We prove that for large times, after a suitable rescaling, the position process resembles a Brownian motion if beta >= 4 + d, a stable process if beta is an element of [d, 4 + d) and an integrated multi-dimensional generalization of a Bessel process if beta is an element of (d - 2, d). The critical cases beta = d, beta =1+ d and beta = 4 + d require special rescalings.